A Derivation of Piketty’s Second Fundamental Law of Capitalism
Notes by Dan Kervick
Several people have asked me for a more mathematically precise formulation of Thomas Piketty’s
Second Fundamental Law of Capitalism, as well as a more careful statement of its proof and of the
boundary conditions over which the proof applies, with some attention to the unusual and exceptional
cases. These notes are an attempt to comply with those requests. While I have attempted to be as
precise as possible in laying out the proof, I have tried to avoid introducing any more rigor than is
necessary for an informed reader to understand how the proof goes through. I have assumed a
familiarity with reasoning about limits at the level of elementary calculus.
Recall that for any year i, the capital-to-income ratio β for that year is defined as:
1. βi= Wi/Yi
where Wi is total accumulated national wealth for year i, and Yi is the national income for year i. Also
recall that if si is the savings rate for that same year, then national wealth grows from year i to the
succeeding year i + 1 according to the rule:
2. Wi+1 = Wi + siYi
Also note that national income changes from one year to the next in accordance with some growth rate
gi for that year that can be defined by:
3. gi = Yi+1/Yi – 1
which gives us the income growth rule:
4. Yi+1 = (1 + gi)Yi
Using these equations, and given the values of these fundamental quantities for year i, we can
determine β for the succeeding year i + 1:
5. βi+1 = (Wi + siYi)/(1 + gi)Yi
which we can also see is equal to:
6. βi+1 = (βi + si)/(1 +gi)
Now let’s look at the evolution of the capital-to-income ratio from year to year, beginning in some
arbitrary year zero:
β0, β1, β2, β3, …
Since si and gi can change from one year to the next, there may be no fixed pattern to this sequence, in
either the short term or the long term. However, we can focus on the special case where the savings
rate and growth rate are fixed constants. In that case, formula 6 contains only a single time-varying
function instead of three time-varying functions, and becomes this simple recursive rule for the
evolution of β from one year to the next:
7. βi+1 = (βi + s)/(1 + g)
Starting with β0 as the capital-to-income ratio for the year 0, we can then work out the capital-toincome ratios for succeeding years 1, 2, 3, etc.
β1 = (β0 + s)/(1 +g)
β2 = [β0 + s + s(1+g)]/(1 + g)2
β3 = [β0 + s + s(1+g) + s(1+g)2]/(1 + g)3
etc.
We can easily prove by mathematical induction the general formula for βn which is evident from the
pattern above:
8. βn = [β0 + s + s(1+g) + s(1+g)2 + … +s(1+g)n-1]/(1 + g)n
Just to be extra clear about what is going on, what we have really done so far is to define a class of
capital-to-income ratio functions over the natural numbers 0, 1, 2, etc., one for each choice of s and g.
When we need to be very specific, we will write such a function in the form βs,g(n). But generally, we will
suppress the subscripts s and g as understood, reduce the argument n to subscript form and just write
βn as we have so far.
Now here are two questions we need to address:
 Under what conditions does a capital-to-income function βs,g(n) converge to a limit?
 When such a function βs,g(n) does converge, to what value does it converge?
Before turning to these questions, let’s enter a few clarifying notes about equation 8. First, the value of
βn is obviously not defined for cases in which g = -1 (apart from the value β0 for the degenerate case of
year 0 alone) since in those cases the denominator of the function is zero. This mathematical fact
conforms with our intuitions, since in any situation in which g = -1, national income immediately goes to
0 in year one and all succeeding years, and there is thus no capital-to-income ratio at all in years 1, 2, 3,
etc. (or if one prefers a standard convention for division by zero, we can add a point-at-infinity to the
real numbers and say the capital-to-income ratio is infinite in those succeeding years).
Also, while the sequence βn is well defined for values of g < -1, such values appear to be economically
meaningless, since national income would in these cases oscillate back and forth annually between
positive and negative values. For example, if the initial national income is $100 and g = -2, then we get
the following sequence for the evolution of national income year-by-year:
$100, -$100, $100, -$100, …
And if g = -3, we get
$100, -$200, $400, -$800, …
A negative national income for some given year is conceptually possible. For example, a society that
produces absolutely nothing in that year, and that makes net positive payments out of its remaining
stock of wealth to people living abroad who have ownership claims on that capital stock, would have
negative income for that year. Nevertheless, such cases are weird and other-worldly, and even weirder
would be a society that had a fixed negative growth rate of less than -1, with the strange oscillations just
described. So while we will return to these cases of g < -1 later, we will do so only for the sake of
mathematical completeness.
A more interesting case occurs if g is negative, but falls between – 1 and 0. This represents a situation of
continually decreasing, but consistently positive, annual income. For example, if the initial national
income is $100 and g = -0.05, then we get the following sequence of national incomes in succeeding
years:
$100, $95, $90.25, $85.74
It is worth looking at such cases, even though these kinds of recessionary negative growth conditions are
generally short-lived in the real world, and don’t have much deep intellectual value where our interest is
in the long-term evolution of β over many years in real-world economies that sustain positive average
growth rates during the period of interest.
Before going on, we need to recall the formula for the sum of the first n terms of a geometric series with
initial term a and geometric ratio r. We can introduce the following notation for this sum:
9. Sa,r(n) = a + ar + ar2 + … + arn-1
The sum is then computed by this formula:
10. Sa,r(n) =
a(1− rn )
1−r
It will be more convenient in what follows to use the equivalent form of this equation that comes from
multiplying both the numerator and denominator by -1.
11. Sa,r(n) =
a(rn −1)
r−1
Note that the sum is not defined by formulas 10 and 11 for the value r = 1. When r = 1, the terms of the
series simply increase incrementally by the addition of a, and we get instead:
12. Sa,1(n) = a · n
This exception will play a small role in what follows.
By applying the notation introduced in the left half of formula 9 to equation 8, which is the formula for
the capital-to-income ratio in year n, we can rewrite equation 8 as:
13. βn = [β0 + Ss,(1+g)(n)]/(1 + g)n
And by applying formula 11 to this result, we get:
14. βn =
s((1+g)n −1)
(1+g)− 1
(1+g)n
β0 +
Progressively reducing this equation, we now obtain:
15. βn =
s((1+g)n −1)
g
(1+g)n
β0 +
and
16. βN =
β0
s((1+g)n −1)
+
n
g(1+g)n
(1+g)
And finally, we get:
17. βn =
β0
(1 + g)n
+
s
g
[1 −
1
]
(1+g)n
This is the equation that we want to examine for its convergence properties as n increases. Recall,
however, that we earlier noted that equations 10 ad 11 are invalid for cases of r = 1, which in this case
corresponds to the case of g = 0. For this special case we have to combine equations 12 and 13 to get:
18. βn = β0 + sn
This sequence clearly does not converge, but increases without bound as n increases. Thus, we get a first
analytic result for the evolution of a capital-to-income ratio function βs,g(n):
19. When g = 0, the capital-to-income ratio function βs,g(n) fails to converge and increases without
bound as n goes from zero to infinity.
Now let’s look at equation 17 for cases in which g > 0. For such cases, the denominator of the two
fractional terms appearing in the addends of equation 17 will increase without bound as n increases,
and thus those fractional terms themselves will approach 0. As a result, βn approaches 0 + s/g (1 – 0),
which is just s/g. So we have:
20. When g > 0, the capital-to-income ratio function converges to s/g as n goes from zero to infinity.
Now let’s consider those cases in which -1 < g < 0, so that annual national income decreases each year at
a fixed annual rate. We could describe such conditions as those of a stable, permanent depression. In
this case, the denominators of the two fractional terms appearing in the addends of equation 17 go
toward zero as n increases, and thus the fractional terms themselves increase without bound. Thus, in
these cases the capital-to-income ratio fails to converge. This makes sense: in such a situation, national
income would be moving toward zero, and national wealth would stop growing appreciably and
approach some limit. So we have our third result.
21. When -1 < g < 0, the capital-to-income ratio function fails to converge and increases without
bound as n goes from zero to infinity.
And we have already noted in passing the following result for the unique case of g = -1:
22. When g = -1, the capital-to-income ratio function βs,g(n) is undefined for all n > 0.
Finally, just for the sake of mathematical completeness, let’s look at the really weird cases: those in
which g < -1, in which case the capital-to-income ratio thus flips back and forth annually from a positive
number to a negative number and a negative number to a positive number. We can divide this
pathological case into three sub-cases, one where -2 < g < -1; one where g = -2, and a third where g < -2.
In the first case, the absolute values of the denominators of the fractional terms approach 0 as n
increases, and so the absolute values of those fractional terms increase without bound (in addition to
their signs oscillating.) Thus, it is clear that:
23. When -2 < g < -1, the capital-to-income ratio function βs,g(n) fails to converge and its absolute
value increases without bound as n goes from zero to infinity.
When g is exactly equal to -2, then the denominators of the fractional terms oscillate back and forth
between 1 and -1. As a result, the first addend oscillates back and forth between -β0 and β0, and the
value of the expression within the square brackets oscillates between 2 and 0, meaning that the second
addend oscillates between 2s/g and 0. So we get:
24. When g = -2, the capital-to-income ratio function βs,g(n) fails to converge and oscillates back and
forth between and -β0 + 2s/g and β0 as n goes from zero to infinity.
Finally, I will leave it to the reader to confirm the following result by inspection:
25. When g < -2, the capital-to-income ratio function βs,g(n) fails to converge, but the absolute value of
the capital-to-income ratio converges to the absolute value of s/g.
Now there are several results here. But I would suggest that the only really important one is proposition
20. The others can all be encapsulated by this proposition:
24. When g ≤ 0, the capital-to-income ratio function βs,g(n) is either undefined for all values of n other
than 0, or fails to converge as n goes from zero to infinity.
I think we are finally in a position to state Piketty’s Second Fundamental law of Capitalism in a clear and
precise form. We can put it like this:
Piketty’s Second Fundamental law of Capitalism:
For any capital-to-income function βs,g(n) with g > 0, 𝐥𝐢𝐦 𝛃𝐬,𝐠 (𝐧) = 𝐬⁄𝐠.
𝐧→∞
When Piketty introduces this law in the text of Capital in the Twenty-First Century, he abbreviates it in
the form β = s/g for the benefit of the general reader. But he then follows up that introductory
statement with several pages of explanatory text and warnings to the reader that make it clear that that
his intent is only to describe the value toward which β would converge in the long run, if s and g remain
relatively constant. Unfortunately, the abbreviated statement of the law has thrown many people off.
Economists are in the habit of using equilibrium methods to turn long-term asymptotic limits of
processes that evolve over time into unchanging constraints on some idealized steady-state equilibrium.
So rather than use the inherently dynamic accounts of wealth, income and the capital-to-income ratio
that are partially explained above, and that Piketty uses to study the dynamical evolution of the
structure of economic inequality as it evolves over time, some commentators have instead gone right to
a steady-state model in which β is set constantly equal to s/g, and from which they then fallaciously
deduce immediate (and absurd) changes in β from the fluctuations in s and g. This is quite far from
Piketty’s intent and misrepresents his framework.
Since part of the point of the Second Fundamental law and its surrounding framework is to analyze what
happens as the capital-to-income ratio increases over time, not what happens in the less usual
conditions when it is stable, the focus on steady-state conditions seems to me misguided. For example,
if a society begins in 2014 with a capital-to-income ratio of 4, a savings rate of 10% and a growth rate of
1.5%, then even if the savings rate and growth rate remained constant indefinitely, it would take a very
long time for β to get near its limiting value of s/g = 6.7. Specifically, β would not get to within 10% of s/g
until the year 2115; it would not get to within 5% of s/g until the year 2159; and it would not get to
within 1% of s/g until the year 2159. The distance between the present value of β and the present value
of s/g is a measure of the present strength of the “force” pushing the capital-to-income ratio higher.
Since some of the forces generating inequality depend on the rapidity with which β is increasing, it is the
strength of this force and the direction in which β is moving that are important. Piketty’s Fundamental
Law should not be interpreted as the statement that β is approximately equal to s/g. Under ordinary
conditions β can be quite far from s/g. That’s part of the point.